| L(s) = 1 | − 3-s − 2·4-s − 2.95·5-s + 1.80·7-s + 9-s − 1.40·11-s + 2·12-s + 1.56·13-s + 2.95·15-s + 4·16-s − 4.88·17-s + 5.65·19-s + 5.91·20-s − 1.80·21-s + 23-s + 3.75·25-s − 27-s − 3.60·28-s + 29-s + 7.71·31-s + 1.40·33-s − 5.32·35-s − 2·36-s + 7.59·37-s − 1.56·39-s − 7.12·41-s − 8.16·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.32·5-s + 0.680·7-s + 0.333·9-s − 0.424·11-s + 0.577·12-s + 0.434·13-s + 0.764·15-s + 16-s − 1.18·17-s + 1.29·19-s + 1.32·20-s − 0.392·21-s + 0.208·23-s + 0.751·25-s − 0.192·27-s − 0.680·28-s + 0.185·29-s + 1.38·31-s + 0.245·33-s − 0.900·35-s − 0.333·36-s + 1.24·37-s − 0.251·39-s − 1.11·41-s − 1.24·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 - 1.80T + 7T^{2} \) |
| 11 | \( 1 + 1.40T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 + 4.88T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 31 | \( 1 - 7.71T + 31T^{2} \) |
| 37 | \( 1 - 7.59T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 + 8.16T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 - 7.51T + 53T^{2} \) |
| 59 | \( 1 + 2.88T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 3.60T + 67T^{2} \) |
| 71 | \( 1 + 1.79T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 - 1.74T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 1.35T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.574543773010552082579151476576, −8.065700589760266060954366627055, −7.40134050165660362669529909184, −6.38568577375406025719048290527, −5.27459370394834878731833024837, −4.65623370356808358009618075925, −4.05982647521062661458735120438, −3.02714477778869143328923114258, −1.19139156176836316175038592211, 0,
1.19139156176836316175038592211, 3.02714477778869143328923114258, 4.05982647521062661458735120438, 4.65623370356808358009618075925, 5.27459370394834878731833024837, 6.38568577375406025719048290527, 7.40134050165660362669529909184, 8.065700589760266060954366627055, 8.574543773010552082579151476576
